In the circuit shown in Figure P8-65, there is an unknown element \(Z_{\mathrm{X}}\). The task is to determine the phasor gain and the output impedance \(Z_{\text {OUT }}\) of the circuit. Before those can be determined, the unknown element needs to be found. If the input impedance was measured to be \(Z_{\mathrm{IN}}=200-j 100 \Omega\) :

(a) Find the unknown impedance \(Z_{\mathrm{X}}\) of the circuit.

(b) Find the output impedance \(Z_{\text {OUT }}\) of the circuit.

(c) Find the phasor gain \(K=\mathbf{V}_{\mathrm{O}} / \mathbf{V}_{\mathrm{S}}\).